R D Sharma Solutions for Chapter: Algebra of Matrices, Exercise 1: EXERCISE

If $A=\left[\begin{array}{rr}3& -5\\ -4& 2\end{array}\right]$, find $A^2-5A-14I$.

If $A=\left[\begin{array}{ll}3& -2\\ 4& -2\end{array}\right]$ , find $k$ such that $A^2=kA-2I_2$.

If $A=\left[\begin{array}{rr}1& 0\\ -1& 7\end{array}\right]$, find $k$ such that  $A^2-8A+kI=O$.

Solve the matrix equation

$\left[\begin{array}{lll}1& 2& 1\end{array}\right]\left[\begin{array}{lll}1& 2& 0\\ 2& 0& 1\\ 1& 0& 2\end{array}\right]\left[\begin{array}{l}0\\ 2\\ x\end{array}\right]=0$

If $A=\left[\begin{array}{rr}3& -5\\ -4& 2\end{array}\right]$ then find $A^2-5A-14I$. Hence, obtain $A^3$.

If $A=\mathrm{diag}\left(a, b, c\right)$, show that $A^n=\mathrm{diag}\left(a^n, b^n, c^n\right)$ for all positive integers $n$.

If $A$ is a square matrix, using mathematical induction prove that ${\left(A^T\right)}^n={\left(A^n\right)}^T$ for all $n\in N$.

Give an example of matrices $A$ and $B$ such that $AB\ne BA$.